LMFDB - Embedded newform 270.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [270,4,Mod(1,270)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(270, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("270.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 270 = 2 \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	270.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$15.9305157015$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	270.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -13.0000 q^{7} -8.00000 q^{8} +O(q^{10})$

\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -13.0000 q^{7} -8.00000 q^{8} -10.0000 q^{10} +30.0000 q^{11} -61.0000 q^{13} +26.0000 q^{14} +16.0000 q^{16} -12.0000 q^{17} -49.0000 q^{19} +20.0000 q^{20} -60.0000 q^{22} -18.0000 q^{23} +25.0000 q^{25} +122.000 q^{26} -52.0000 q^{28} +186.000 q^{29} -160.000 q^{31} -32.0000 q^{32} +24.0000 q^{34} -65.0000 q^{35} -91.0000 q^{37} +98.0000 q^{38} -40.0000 q^{40} -378.000 q^{41} -268.000 q^{43} +120.000 q^{44} +36.0000 q^{46} -144.000 q^{47} -174.000 q^{49} -50.0000 q^{50} -244.000 q^{52} -570.000 q^{53} +150.000 q^{55} +104.000 q^{56} -372.000 q^{58} -204.000 q^{59} -877.000 q^{61} +320.000 q^{62} +64.0000 q^{64} -305.000 q^{65} -187.000 q^{67} -48.0000 q^{68} +130.000 q^{70} +606.000 q^{71} +431.000 q^{73} +182.000 q^{74} -196.000 q^{76} -390.000 q^{77} +1151.00 q^{79} +80.0000 q^{80} +756.000 q^{82} -102.000 q^{83} -60.0000 q^{85} +536.000 q^{86} -240.000 q^{88} -984.000 q^{89} +793.000 q^{91} -72.0000 q^{92} +288.000 q^{94} -245.000 q^{95} -265.000 q^{97} +348.000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−13.0000	−0.701934	−0.350967	−	0.936388i	$-0.614147\pi$
$7$	−13.0000	−0.701934	−0.350967	+	0.936388i	$0.614147\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−10.0000	−0.316228
$11$	30.0000	0.822304	0.411152	−	0.911567i	$-0.365127\pi$
$11$	30.0000	0.822304	0.411152	+	0.911567i	$0.365127\pi$
$12$	0	0
$13$	−61.0000	−1.30141	−0.650706	−	0.759330i	$-0.725527\pi$
$13$	−61.0000	−1.30141	−0.650706	+	0.759330i	$0.725527\pi$
$14$	26.0000	0.496342
$15$	0	0
$16$	16.0000	0.250000
$17$	−12.0000	−0.171202	−0.0856008	−	0.996330i	$-0.527281\pi$
$17$	−12.0000	−0.171202	−0.0856008	+	0.996330i	$0.527281\pi$
$18$	0	0
$19$	−49.0000	−0.591651	−0.295826	−	0.955242i	$-0.595595\pi$
$19$	−49.0000	−0.591651	−0.295826	+	0.955242i	$0.595595\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	−60.0000	−0.581456
$23$	−18.0000	−0.163185	−0.0815926	−	0.996666i	$-0.526001\pi$
$23$	−18.0000	−0.163185	−0.0815926	+	0.996666i	$0.526001\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	122.000	0.920237
$27$	0	0
$28$	−52.0000	−0.350967
$29$	186.000	1.19101	0.595506	−	0.803351i	$-0.296952\pi$
$29$	186.000	1.19101	0.595506	+	0.803351i	$0.296952\pi$
$30$	0	0
$31$	−160.000	−0.926995	−0.463498	−	0.886098i	$-0.653406\pi$
$31$	−160.000	−0.926995	−0.463498	+	0.886098i	$0.653406\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	24.0000	0.121058
$35$	−65.0000	−0.313914
$36$	0	0
$37$	−91.0000	−0.404333	−0.202166	−	0.979351i	$-0.564798\pi$
$37$	−91.0000	−0.404333	−0.202166	+	0.979351i	$0.564798\pi$
$38$	98.0000	0.418361
$39$	0	0
$40$	−40.0000	−0.158114
$41$	−378.000	−1.43985	−0.719923	−	0.694054i	$-0.755823\pi$
$41$	−378.000	−1.43985	−0.719923	+	0.694054i	$0.755823\pi$
$42$	0	0
$43$	−268.000	−0.950456	−0.475228	−	0.879863i	$-0.657634\pi$
$43$	−268.000	−0.950456	−0.475228	+	0.879863i	$0.657634\pi$
$44$	120.000	0.411152
$45$	0	0
$46$	36.0000	0.115389
$47$	−144.000	−0.446906	−0.223453	−	0.974715i	$-0.571733\pi$
$47$	−144.000	−0.446906	−0.223453	+	0.974715i	$0.571733\pi$
$48$	0	0
$49$	−174.000	−0.507289
$50$	−50.0000	−0.141421
$51$	0	0
$52$	−244.000	−0.650706
$53$	−570.000	−1.47727	−0.738637	−	0.674103i	$-0.764530\pi$
$53$	−570.000	−1.47727	−0.738637	+	0.674103i	$0.764530\pi$
$54$	0	0
$55$	150.000	0.367745
$56$	104.000	0.248171
$57$	0	0
$58$	−372.000	−0.842172
$59$	−204.000	−0.450145	−0.225072	−	0.974342i	$-0.572262\pi$
$59$	−204.000	−0.450145	−0.225072	+	0.974342i	$0.572262\pi$
$60$	0	0
$61$	−877.000	−1.84079	−0.920396	−	0.390987i	$-0.872134\pi$
$61$	−877.000	−1.84079	−0.920396	+	0.390987i	$0.872134\pi$
$62$	320.000	0.655485
$63$	0	0
$64$	64.0000	0.125000
$65$	−305.000	−0.582009
$66$	0	0
$67$	−187.000	−0.340980	−0.170490	−	0.985359i	$-0.554535\pi$
$67$	−187.000	−0.340980	−0.170490	+	0.985359i	$0.554535\pi$
$68$	−48.0000	−0.0856008
$69$	0	0
$70$	130.000	0.221971
$71$	606.000	1.01294	0.506472	−	0.862257i	$-0.330950\pi$
$71$	606.000	1.01294	0.506472	+	0.862257i	$0.330950\pi$
$72$	0	0
$73$	431.000	0.691024	0.345512	−	0.938414i	$-0.387705\pi$
$73$	431.000	0.691024	0.345512	+	0.938414i	$0.387705\pi$
$74$	182.000	0.285906
$75$	0	0
$76$	−196.000	−0.295826
$77$	−390.000	−0.577203
$78$	0	0
$79$	1151.00	1.63921	0.819605	−	0.572929i	$-0.194193\pi$
$79$	1151.00	1.63921	0.819605	+	0.572929i	$0.194193\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	756.000	1.01812
$83$	−102.000	−0.134891	−0.0674455	−	0.997723i	$-0.521485\pi$
$83$	−102.000	−0.134891	−0.0674455	+	0.997723i	$0.521485\pi$
$84$	0	0
$85$	−60.0000	−0.0765637
$86$	536.000	0.672074
$87$	0	0
$88$	−240.000	−0.290728
$89$	−984.000	−1.17195	−0.585976	−	0.810328i	$-0.699289\pi$
$89$	−984.000	−1.17195	−0.585976	+	0.810328i	$0.699289\pi$
$90$	0	0
$91$	793.000	0.913505
$92$	−72.0000	−0.0815926
$93$	0	0
$94$	288.000	0.316010
$95$	−245.000	−0.264594
$96$	0	0
$97$	−265.000	−0.277388	−0.138694	−	0.990335i	$-0.544291\pi$
$97$	−265.000	−0.277388	−0.138694	+	0.990335i	$0.544291\pi$
$98$	348.000	0.358707
$99$	0	0
$100$	100.000	0.100000
$101$	1248.00	1.22951	0.614756	−	0.788718i	$-0.289255\pi$
$101$	1248.00	1.22951	0.614756	+	0.788718i	$0.289255\pi$
$102$	0	0
$103$	−1225.00	−1.17187	−0.585936	−	0.810357i	$-0.699273\pi$
$103$	−1225.00	−1.17187	−0.585936	+	0.810357i	$0.699273\pi$
$104$	488.000	0.460119
$105$	0	0
$106$	1140.00	1.04459
$107$	78.0000	0.0704724	0.0352362	−	0.999379i	$-0.488782\pi$
$107$	78.0000	0.0704724	0.0352362	+	0.999379i	$0.488782\pi$
$108$	0	0
$109$	2198.00	1.93147	0.965735	−	0.259530i	$-0.0835678\pi$
$109$	2198.00	1.93147	0.965735	+	0.259530i	$0.0835678\pi$
$110$	−300.000	−0.260035
$111$	0	0
$112$	−208.000	−0.175484
$113$	−1986.00	−1.65334	−0.826669	−	0.562689i	$-0.809767\pi$
$113$	−1986.00	−1.65334	−0.826669	+	0.562689i	$0.809767\pi$
$114$	0	0
$115$	−90.0000	−0.0729786
$116$	744.000	0.595506
$117$	0	0
$118$	408.000	0.318300
$119$	156.000	0.120172
$120$	0	0
$121$	−431.000	−0.323817
$122$	1754.00	1.30164
$123$	0	0
$124$	−640.000	−0.463498
$125$	125.000	0.0894427
$126$	0	0
$127$	2792.00	1.95079	0.975393	−	0.220471i	$-0.0707595\pi$
$127$	2792.00	1.95079	0.975393	+	0.220471i	$0.0707595\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	610.000	0.411543
$131$	708.000	0.472200	0.236100	−	0.971729i	$-0.424131\pi$
$131$	708.000	0.472200	0.236100	+	0.971729i	$0.424131\pi$
$132$	0	0
$133$	637.000	0.415300
$134$	374.000	0.241110
$135$	0	0
$136$	96.0000	0.0605289
$137$	1686.00	1.05142	0.525711	−	0.850663i	$-0.323799\pi$
$137$	1686.00	1.05142	0.525711	+	0.850663i	$0.323799\pi$
$138$	0	0
$139$	−307.000	−0.187334	−0.0936669	−	0.995604i	$-0.529859\pi$
$139$	−307.000	−0.187334	−0.0936669	+	0.995604i	$0.529859\pi$
$140$	−260.000	−0.156957
$141$	0	0
$142$	−1212.00	−0.716259
$143$	−1830.00	−1.07016
$144$	0	0
$145$	930.000	0.532637
$146$	−862.000	−0.488628
$147$	0	0
$148$	−364.000	−0.202166
$149$	1812.00	0.996274	0.498137	−	0.867098i	$-0.334018\pi$
$149$	1812.00	0.996274	0.498137	+	0.867098i	$0.334018\pi$
$150$	0	0
$151$	203.000	0.109403	0.0547017	−	0.998503i	$-0.482579\pi$
$151$	203.000	0.109403	0.0547017	+	0.998503i	$0.482579\pi$
$152$	392.000	0.209180
$153$	0	0
$154$	780.000	0.408144
$155$	−800.000	−0.414565
$156$	0	0
$157$	−214.000	−0.108784	−0.0543919	−	0.998520i	$-0.517322\pi$
$157$	−214.000	−0.108784	−0.0543919	+	0.998520i	$0.517322\pi$
$158$	−2302.00	−1.15910
$159$	0	0
$160$	−160.000	−0.0790569
$161$	234.000	0.114545
$162$	0	0
$163$	−673.000	−0.323395	−0.161698	−	0.986840i	$-0.551697\pi$
$163$	−673.000	−0.323395	−0.161698	+	0.986840i	$0.551697\pi$
$164$	−1512.00	−0.719923
$165$	0	0
$166$	204.000	0.0953824
$167$	3696.00	1.71261	0.856303	−	0.516474i	$-0.172756\pi$
$167$	3696.00	1.71261	0.856303	+	0.516474i	$0.172756\pi$
$168$	0	0
$169$	1524.00	0.693673
$170$	120.000	0.0541387
$171$	0	0
$172$	−1072.00	−0.475228
$173$	−3132.00	−1.37643	−0.688213	−	0.725509i	$-0.741604\pi$
$173$	−3132.00	−1.37643	−0.688213	+	0.725509i	$0.741604\pi$
$174$	0	0
$175$	−325.000	−0.140387
$176$	480.000	0.205576
$177$	0	0
$178$	1968.00	0.828696
$179$	−510.000	−0.212956	−0.106478	−	0.994315i	$-0.533957\pi$
$179$	−510.000	−0.212956	−0.106478	+	0.994315i	$0.533957\pi$
$180$	0	0
$181$	−1087.00	−0.446387	−0.223194	−	0.974774i	$-0.571648\pi$
$181$	−1087.00	−0.446387	−0.223194	+	0.974774i	$0.571648\pi$
$182$	−1586.00	−0.645946
$183$	0	0
$184$	144.000	0.0576947
$185$	−455.000	−0.180823
$186$	0	0
$187$	−360.000	−0.140780
$188$	−576.000	−0.223453
$189$	0	0
$190$	490.000	0.187097
$191$	4056.00	1.53655	0.768277	−	0.640117i	$-0.221114\pi$
$191$	4056.00	1.53655	0.768277	+	0.640117i	$0.221114\pi$
$192$	0	0
$193$	473.000	0.176411	0.0882054	−	0.996102i	$-0.471887\pi$
$193$	473.000	0.176411	0.0882054	+	0.996102i	$0.471887\pi$
$194$	530.000	0.196143
$195$	0	0
$196$	−696.000	−0.253644
$197$	−2556.00	−0.924403	−0.462202	−	0.886775i	$-0.652940\pi$
$197$	−2556.00	−0.924403	−0.462202	+	0.886775i	$0.652940\pi$
$198$	0	0
$199$	−2923.00	−1.04124	−0.520618	−	0.853790i	$-0.674298\pi$
$199$	−2923.00	−1.04124	−0.520618	+	0.853790i	$0.674298\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	−2496.00	−0.869396
$203$	−2418.00	−0.836011
$204$	0	0
$205$	−1890.00	−0.643919
$206$	2450.00	0.828639
$207$	0	0
$208$	−976.000	−0.325353
$209$	−1470.00	−0.486517
$210$	0	0
$211$	−3175.00	−1.03591	−0.517953	−	0.855409i	$-0.673306\pi$
$211$	−3175.00	−1.03591	−0.517953	+	0.855409i	$0.673306\pi$
$212$	−2280.00	−0.738637
$213$	0	0
$214$	−156.000	−0.0498315
$215$	−1340.00	−0.425057
$216$	0	0
$217$	2080.00	0.650689
$218$	−4396.00	−1.36576
$219$	0	0
$220$	600.000	0.183873
$221$	732.000	0.222804
$222$	0	0
$223$	−2176.00	−0.653434	−0.326717	−	0.945122i	$-0.605942\pi$
$223$	−2176.00	−0.653434	−0.326717	+	0.945122i	$0.605942\pi$
$224$	416.000	0.124086
$225$	0	0
$226$	3972.00	1.16909
$227$	3834.00	1.12102	0.560510	−	0.828148i	$-0.310605\pi$
$227$	3834.00	1.12102	0.560510	+	0.828148i	$0.310605\pi$
$228$	0	0
$229$	−3202.00	−0.923992	−0.461996	−	0.886882i	$-0.652867\pi$
$229$	−3202.00	−0.923992	−0.461996	+	0.886882i	$0.652867\pi$
$230$	180.000	0.0516037
$231$	0	0
$232$	−1488.00	−0.421086
$233$	−4152.00	−1.16741	−0.583705	−	0.811966i	$-0.698398\pi$
$233$	−4152.00	−1.16741	−0.583705	+	0.811966i	$0.698398\pi$
$234$	0	0
$235$	−720.000	−0.199862
$236$	−816.000	−0.225072
$237$	0	0
$238$	−312.000	−0.0849746
$239$	5466.00	1.47936	0.739678	−	0.672961i	$-0.234978\pi$
$239$	5466.00	1.47936	0.739678	+	0.672961i	$0.234978\pi$
$240$	0	0
$241$	−943.000	−0.252050	−0.126025	−	0.992027i	$-0.540222\pi$
$241$	−943.000	−0.252050	−0.126025	+	0.992027i	$0.540222\pi$
$242$	862.000	0.228973
$243$	0	0
$244$	−3508.00	−0.920396
$245$	−870.000	−0.226866
$246$	0	0
$247$	2989.00	0.769982
$248$	1280.00	0.327742
$249$	0	0
$250$	−250.000	−0.0632456
$251$	7290.00	1.83323	0.916615	−	0.399771i	$-0.130910\pi$
$251$	7290.00	1.83323	0.916615	+	0.399771i	$0.130910\pi$
$252$	0	0
$253$	−540.000	−0.134188
$254$	−5584.00	−1.37941
$255$	0	0
$256$	256.000	0.0625000
$257$	−312.000	−0.0757277	−0.0378639	−	0.999283i	$-0.512055\pi$
$257$	−312.000	−0.0757277	−0.0378639	+	0.999283i	$0.512055\pi$
$258$	0	0
$259$	1183.00	0.283815
$260$	−1220.00	−0.291005
$261$	0	0
$262$	−1416.00	−0.333896
$263$	8004.00	1.87661	0.938304	−	0.345812i	$-0.112397\pi$
$263$	8004.00	1.87661	0.938304	+	0.345812i	$0.112397\pi$
$264$	0	0
$265$	−2850.00	−0.660657
$266$	−1274.00	−0.293661
$267$	0	0
$268$	−748.000	−0.170490
$269$	324.000	0.0734373	0.0367186	−	0.999326i	$-0.488309\pi$
$269$	324.000	0.0734373	0.0367186	+	0.999326i	$0.488309\pi$
$270$	0	0
$271$	−7849.00	−1.75938	−0.879692	−	0.475545i	$-0.842251\pi$
$271$	−7849.00	−1.75938	−0.879692	+	0.475545i	$0.842251\pi$
$272$	−192.000	−0.0428004
$273$	0	0
$274$	−3372.00	−0.743467
$275$	750.000	0.164461
$276$	0	0
$277$	−5758.00	−1.24897	−0.624485	−	0.781037i	$-0.714691\pi$
$277$	−5758.00	−1.24897	−0.624485	+	0.781037i	$0.714691\pi$
$278$	614.000	0.132465
$279$	0	0
$280$	520.000	0.110986
$281$	2688.00	0.570650	0.285325	−	0.958431i	$-0.407898\pi$
$281$	2688.00	0.570650	0.285325	+	0.958431i	$0.407898\pi$
$282$	0	0
$283$	3260.00	0.684759	0.342380	−	0.939562i	$-0.388767\pi$
$283$	3260.00	0.684759	0.342380	+	0.939562i	$0.388767\pi$
$284$	2424.00	0.506472
$285$	0	0
$286$	3660.00	0.756714
$287$	4914.00	1.01068
$288$	0	0
$289$	−4769.00	−0.970690
$290$	−1860.00	−0.376631
$291$	0	0
$292$	1724.00	0.345512
$293$	−5922.00	−1.18077	−0.590387	−	0.807120i	$-0.701025\pi$
$293$	−5922.00	−1.18077	−0.590387	+	0.807120i	$0.701025\pi$
$294$	0	0
$295$	−1020.00	−0.201311
$296$	728.000	0.142953
$297$	0	0
$298$	−3624.00	−0.704472
$299$	1098.00	0.212371
$300$	0	0
$301$	3484.00	0.667158
$302$	−406.000	−0.0773599
$303$	0	0
$304$	−784.000	−0.147913
$305$	−4385.00	−0.823227
$306$	0	0
$307$	3728.00	0.693056	0.346528	−	0.938040i	$-0.387361\pi$
$307$	3728.00	0.693056	0.346528	+	0.938040i	$0.387361\pi$
$308$	−1560.00	−0.288601
$309$	0	0
$310$	1600.00	0.293142
$311$	732.000	0.133466	0.0667330	−	0.997771i	$-0.478742\pi$
$311$	732.000	0.133466	0.0667330	+	0.997771i	$0.478742\pi$
$312$	0	0
$313$	5357.00	0.967398	0.483699	−	0.875234i	$-0.339293\pi$
$313$	5357.00	0.967398	0.483699	+	0.875234i	$0.339293\pi$
$314$	428.000	0.0769218
$315$	0	0
$316$	4604.00	0.819605
$317$	4572.00	0.810060	0.405030	−	0.914303i	$-0.367261\pi$
$317$	4572.00	0.810060	0.405030	+	0.914303i	$0.367261\pi$
$318$	0	0
$319$	5580.00	0.979373
$320$	320.000	0.0559017
$321$	0	0
$322$	−468.000	−0.0809957
$323$	588.000	0.101292
$324$	0	0
$325$	−1525.00	−0.260282
$326$	1346.00	0.228675
$327$	0	0
$328$	3024.00	0.509062
$329$	1872.00	0.313698
$330$	0	0
$331$	845.000	0.140318	0.0701592	−	0.997536i	$-0.477649\pi$
$331$	845.000	0.140318	0.0701592	+	0.997536i	$0.477649\pi$
$332$	−408.000	−0.0674455
$333$	0	0
$334$	−7392.00	−1.21099
$335$	−935.000	−0.152491
$336$	0	0
$337$	8723.00	1.41001	0.705003	−	0.709204i	$-0.250946\pi$
$337$	8723.00	1.41001	0.705003	+	0.709204i	$0.250946\pi$
$338$	−3048.00	−0.490501
$339$	0	0
$340$	−240.000	−0.0382818
$341$	−4800.00	−0.762271
$342$	0	0
$343$	6721.00	1.05802
$344$	2144.00	0.336037
$345$	0	0
$346$	6264.00	0.973280
$347$	−9018.00	−1.39513	−0.697567	−	0.716519i	$-0.745734\pi$
$347$	−9018.00	−1.39513	−0.697567	+	0.716519i	$0.745734\pi$
$348$	0	0
$349$	5759.00	0.883301	0.441651	−	0.897187i	$-0.354393\pi$
$349$	5759.00	0.883301	0.441651	+	0.897187i	$0.354393\pi$
$350$	650.000	0.0992685
$351$	0	0
$352$	−960.000	−0.145364
$353$	5772.00	0.870291	0.435145	−	0.900360i	$-0.356697\pi$
$353$	5772.00	0.870291	0.435145	+	0.900360i	$0.356697\pi$
$354$	0	0
$355$	3030.00	0.453002
$356$	−3936.00	−0.585976
$357$	0	0
$358$	1020.00	0.150583
$359$	2046.00	0.300790	0.150395	−	0.988626i	$-0.451945\pi$
$359$	2046.00	0.300790	0.150395	+	0.988626i	$0.451945\pi$
$360$	0	0
$361$	−4458.00	−0.649949
$362$	2174.00	0.315643
$363$	0	0
$364$	3172.00	0.456753
$365$	2155.00	0.309035
$366$	0	0
$367$	−1069.00	−0.152047	−0.0760236	−	0.997106i	$-0.524222\pi$
$367$	−1069.00	−0.152047	−0.0760236	+	0.997106i	$0.524222\pi$
$368$	−288.000	−0.0407963
$369$	0	0
$370$	910.000	0.127861
$371$	7410.00	1.03695
$372$	0	0
$373$	7133.00	0.990168	0.495084	−	0.868845i	$-0.335137\pi$
$373$	7133.00	0.990168	0.495084	+	0.868845i	$0.335137\pi$
$374$	720.000	0.0995463
$375$	0	0
$376$	1152.00	0.158005
$377$	−11346.0	−1.55000
$378$	0	0
$379$	−8557.00	−1.15975	−0.579873	−	0.814707i	$-0.696898\pi$
$379$	−8557.00	−1.15975	−0.579873	+	0.814707i	$0.696898\pi$
$380$	−980.000	−0.132297
$381$	0	0
$382$	−8112.00	−1.08651
$383$	−14328.0	−1.91156	−0.955779	−	0.294086i	$-0.904985\pi$
$383$	−14328.0	−1.91156	−0.955779	+	0.294086i	$0.904985\pi$
$384$	0	0
$385$	−1950.00	−0.258133
$386$	−946.000	−0.124741
$387$	0	0
$388$	−1060.00	−0.138694
$389$	−13500.0	−1.75958	−0.879791	−	0.475361i	$-0.842317\pi$
$389$	−13500.0	−1.75958	−0.879791	+	0.475361i	$0.842317\pi$
$390$	0	0
$391$	216.000	0.0279376
$392$	1392.00	0.179354
$393$	0	0
$394$	5112.00	0.653652
$395$	5755.00	0.733077
$396$	0	0
$397$	1334.00	0.168644	0.0843218	−	0.996439i	$-0.473128\pi$
$397$	1334.00	0.168644	0.0843218	+	0.996439i	$0.473128\pi$
$398$	5846.00	0.736265
$399$	0	0
$400$	400.000	0.0500000
$401$	−3474.00	−0.432627	−0.216313	−	0.976324i	$-0.569403\pi$
$401$	−3474.00	−0.432627	−0.216313	+	0.976324i	$0.569403\pi$
$402$	0	0
$403$	9760.00	1.20640
$404$	4992.00	0.614756
$405$	0	0
$406$	4836.00	0.591149
$407$	−2730.00	−0.332484
$408$	0	0
$409$	569.000	0.0687903	0.0343952	−	0.999408i	$-0.489050\pi$
$409$	569.000	0.0687903	0.0343952	+	0.999408i	$0.489050\pi$
$410$	3780.00	0.455319
$411$	0	0
$412$	−4900.00	−0.585936
$413$	2652.00	0.315972
$414$	0	0
$415$	−510.000	−0.0603251
$416$	1952.00	0.230059
$417$	0	0
$418$	2940.00	0.344019
$419$	9132.00	1.06474	0.532372	−	0.846511i	$-0.321301\pi$
$419$	9132.00	1.06474	0.532372	+	0.846511i	$0.321301\pi$
$420$	0	0
$421$	−2971.00	−0.343937	−0.171969	−	0.985102i	$-0.555013\pi$
$421$	−2971.00	−0.343937	−0.171969	+	0.985102i	$0.555013\pi$
$422$	6350.00	0.732496
$423$	0	0
$424$	4560.00	0.522295
$425$	−300.000	−0.0342403
$426$	0	0
$427$	11401.0	1.29211
$428$	312.000	0.0352362
$429$	0	0
$430$	2680.00	0.300561
$431$	−12042.0	−1.34581	−0.672903	−	0.739730i	$-0.734953\pi$
$431$	−12042.0	−1.34581	−0.672903	+	0.739730i	$0.734953\pi$
$432$	0	0
$433$	−8566.00	−0.950706	−0.475353	−	0.879795i	$-0.657680\pi$
$433$	−8566.00	−0.950706	−0.475353	+	0.879795i	$0.657680\pi$
$434$	−4160.00	−0.460107
$435$	0	0
$436$	8792.00	0.965735
$437$	882.000	0.0965487
$438$	0	0
$439$	7400.00	0.804516	0.402258	−	0.915526i	$-0.368225\pi$
$439$	7400.00	0.804516	0.402258	+	0.915526i	$0.368225\pi$
$440$	−1200.00	−0.130018
$441$	0	0
$442$	−1464.00	−0.157546
$443$	2580.00	0.276703	0.138352	−	0.990383i	$-0.455820\pi$
$443$	2580.00	0.276703	0.138352	+	0.990383i	$0.455820\pi$
$444$	0	0
$445$	−4920.00	−0.524113
$446$	4352.00	0.462047
$447$	0	0
$448$	−832.000	−0.0877418
$449$	−13200.0	−1.38741	−0.693704	−	0.720260i	$-0.744023\pi$
$449$	−13200.0	−1.38741	−0.693704	+	0.720260i	$0.744023\pi$
$450$	0	0
$451$	−11340.0	−1.18399
$452$	−7944.00	−0.826669
$453$	0	0
$454$	−7668.00	−0.792681
$455$	3965.00	0.408532
$456$	0	0
$457$	18038.0	1.84635	0.923175	−	0.384380i	$-0.125585\pi$
$457$	18038.0	1.84635	0.923175	+	0.384380i	$0.125585\pi$
$458$	6404.00	0.653361
$459$	0	0
$460$	−360.000	−0.0364893
$461$	−5544.00	−0.560108	−0.280054	−	0.959984i	$-0.590352\pi$
$461$	−5544.00	−0.560108	−0.280054	+	0.959984i	$0.590352\pi$
$462$	0	0
$463$	−17137.0	−1.72014	−0.860069	−	0.510178i	$-0.829580\pi$
$463$	−17137.0	−1.72014	−0.860069	+	0.510178i	$0.829580\pi$
$464$	2976.00	0.297753
$465$	0	0
$466$	8304.00	0.825484
$467$	15888.0	1.57432	0.787162	−	0.616747i	$-0.211550\pi$
$467$	15888.0	1.57432	0.787162	+	0.616747i	$0.211550\pi$
$468$	0	0
$469$	2431.00	0.239346
$470$	1440.00	0.141324
$471$	0	0
$472$	1632.00	0.159150
$473$	−8040.00	−0.781564
$474$	0	0
$475$	−1225.00	−0.118330
$476$	624.000	0.0600861
$477$	0	0
$478$	−10932.0	−1.04606
$479$	−3942.00	−0.376022	−0.188011	−	0.982167i	$-0.560204\pi$
$479$	−3942.00	−0.376022	−0.188011	+	0.982167i	$0.560204\pi$
$480$	0	0
$481$	5551.00	0.526203
$482$	1886.00	0.178226
$483$	0	0
$484$	−1724.00	−0.161908
$485$	−1325.00	−0.124052
$486$	0	0
$487$	1379.00	0.128313	0.0641565	−	0.997940i	$-0.479564\pi$
$487$	1379.00	0.128313	0.0641565	+	0.997940i	$0.479564\pi$
$488$	7016.00	0.650818
$489$	0	0
$490$	1740.00	0.160419
$491$	−14214.0	−1.30645	−0.653227	−	0.757162i	$-0.726585\pi$
$491$	−14214.0	−1.30645	−0.653227	+	0.757162i	$0.726585\pi$
$492$	0	0
$493$	−2232.00	−0.203903
$494$	−5978.00	−0.544459
$495$	0	0
$496$	−2560.00	−0.231749
$497$	−7878.00	−0.711019
$498$	0	0
$499$	9992.00	0.896400	0.448200	−	0.893933i	$-0.352065\pi$
$499$	9992.00	0.896400	0.448200	+	0.893933i	$0.352065\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	−14580.0	−1.29629
$503$	21258.0	1.88439	0.942194	−	0.335067i	$-0.108759\pi$
$503$	21258.0	1.88439	0.942194	+	0.335067i	$0.108759\pi$
$504$	0	0
$505$	6240.00	0.549854
$506$	1080.00	0.0948851
$507$	0	0
$508$	11168.0	0.975393
$509$	16614.0	1.44676	0.723382	−	0.690448i	$-0.242587\pi$
$509$	16614.0	1.44676	0.723382	+	0.690448i	$0.242587\pi$
$510$	0	0
$511$	−5603.00	−0.485053
$512$	−512.000	−0.0441942
$513$	0	0
$514$	624.000	0.0535476
$515$	−6125.00	−0.524077
$516$	0	0
$517$	−4320.00	−0.367492
$518$	−2366.00	−0.200687
$519$	0	0
$520$	2440.00	0.205771
$521$	−11838.0	−0.995455	−0.497728	−	0.867333i	$-0.665832\pi$
$521$	−11838.0	−0.995455	−0.497728	+	0.867333i	$0.665832\pi$
$522$	0	0
$523$	2201.00	0.184021	0.0920105	−	0.995758i	$-0.470671\pi$
$523$	2201.00	0.184021	0.0920105	+	0.995758i	$0.470671\pi$
$524$	2832.00	0.236100
$525$	0	0
$526$	−16008.0	−1.32696
$527$	1920.00	0.158703
$528$	0	0
$529$	−11843.0	−0.973371
$530$	5700.00	0.467155
$531$	0	0
$532$	2548.00	0.207650
$533$	23058.0	1.87383
$534$	0	0
$535$	390.000	0.0315162
$536$	1496.00	0.120555
$537$	0	0
$538$	−648.000	−0.0519280
$539$	−5220.00	−0.417145
$540$	0	0
$541$	−10795.0	−0.857880	−0.428940	−	0.903333i	$-0.641113\pi$
$541$	−10795.0	−0.857880	−0.428940	+	0.903333i	$0.641113\pi$
$542$	15698.0	1.24407
$543$	0	0
$544$	384.000	0.0302645
$545$	10990.0	0.863780
$546$	0	0
$547$	−14185.0	−1.10879	−0.554394	−	0.832254i	$-0.687050\pi$
$547$	−14185.0	−1.10879	−0.554394	+	0.832254i	$0.687050\pi$
$548$	6744.00	0.525711
$549$	0	0
$550$	−1500.00	−0.116291
$551$	−9114.00	−0.704663
$552$	0	0
$553$	−14963.0	−1.15062
$554$	11516.0	0.883155
$555$	0	0
$556$	−1228.00	−0.0936669
$557$	3576.00	0.272029	0.136014	−	0.990707i	$-0.456571\pi$
$557$	3576.00	0.272029	0.136014	+	0.990707i	$0.456571\pi$
$558$	0	0
$559$	16348.0	1.23694
$560$	−1040.00	−0.0784786
$561$	0	0
$562$	−5376.00	−0.403510
$563$	9132.00	0.683602	0.341801	−	0.939772i	$-0.388963\pi$
$563$	9132.00	0.683602	0.341801	+	0.939772i	$0.388963\pi$
$564$	0	0
$565$	−9930.00	−0.739395
$566$	−6520.00	−0.484198
$567$	0	0
$568$	−4848.00	−0.358130
$569$	−25020.0	−1.84340	−0.921699	−	0.387907i	$-0.873198\pi$
$569$	−25020.0	−1.84340	−0.921699	+	0.387907i	$0.873198\pi$
$570$	0	0
$571$	−15997.0	−1.17242	−0.586212	−	0.810158i	$-0.699381\pi$
$571$	−15997.0	−1.17242	−0.586212	+	0.810158i	$0.699381\pi$
$572$	−7320.00	−0.535078
$573$	0	0
$574$	−9828.00	−0.714656
$575$	−450.000	−0.0326370
$576$	0	0
$577$	−5971.00	−0.430808	−0.215404	−	0.976525i	$-0.569107\pi$
$577$	−5971.00	−0.430808	−0.215404	+	0.976525i	$0.569107\pi$
$578$	9538.00	0.686381
$579$	0	0
$580$	3720.00	0.266318
$581$	1326.00	0.0946846
$582$	0	0
$583$	−17100.0	−1.21477
$584$	−3448.00	−0.244314
$585$	0	0
$586$	11844.0	0.834934
$587$	−19242.0	−1.35299	−0.676493	−	0.736449i	$-0.736501\pi$
$587$	−19242.0	−1.35299	−0.676493	+	0.736449i	$0.736501\pi$
$588$	0	0
$589$	7840.00	0.548458
$590$	2040.00	0.142348
$591$	0	0
$592$	−1456.00	−0.101083
$593$	−8118.00	−0.562169	−0.281085	−	0.959683i	$-0.590694\pi$
$593$	−8118.00	−0.562169	−0.281085	+	0.959683i	$0.590694\pi$
$594$	0	0
$595$	780.000	0.0537427
$596$	7248.00	0.498137
$597$	0	0
$598$	−2196.00	−0.150169
$599$	−1902.00	−0.129739	−0.0648695	−	0.997894i	$-0.520663\pi$
$599$	−1902.00	−0.129739	−0.0648695	+	0.997894i	$0.520663\pi$
$600$	0	0
$601$	−14074.0	−0.955225	−0.477613	−	0.878571i	$-0.658498\pi$
$601$	−14074.0	−0.955225	−0.477613	+	0.878571i	$0.658498\pi$
$602$	−6968.00	−0.471752
$603$	0	0
$604$	812.000	0.0547017
$605$	−2155.00	−0.144815
$606$	0	0
$607$	−13825.0	−0.924447	−0.462224	−	0.886763i	$-0.652948\pi$
$607$	−13825.0	−0.924447	−0.462224	+	0.886763i	$0.652948\pi$
$608$	1568.00	0.104590
$609$	0	0
$610$	8770.00	0.582110
$611$	8784.00	0.581608
$612$	0	0
$613$	15569.0	1.02582	0.512909	−	0.858443i	$-0.328568\pi$
$613$	15569.0	1.02582	0.512909	+	0.858443i	$0.328568\pi$
$614$	−7456.00	−0.490065
$615$	0	0
$616$	3120.00	0.204072
$617$	−11922.0	−0.777896	−0.388948	−	0.921260i	$-0.627161\pi$
$617$	−11922.0	−0.777896	−0.388948	+	0.921260i	$0.627161\pi$
$618$	0	0
$619$	6899.00	0.447971	0.223986	−	0.974592i	$-0.428093\pi$
$619$	6899.00	0.447971	0.223986	+	0.974592i	$0.428093\pi$
$620$	−3200.00	−0.207282
$621$	0	0
$622$	−1464.00	−0.0943747
$623$	12792.0	0.822633
$624$	0	0
$625$	625.000	0.0400000
$626$	−10714.0	−0.684054
$627$	0	0
$628$	−856.000	−0.0543919
$629$	1092.00	0.0692224
$630$	0	0
$631$	11711.0	0.738839	0.369420	−	0.929263i	$-0.379557\pi$
$631$	11711.0	0.738839	0.369420	+	0.929263i	$0.379557\pi$
$632$	−9208.00	−0.579548
$633$	0	0
$634$	−9144.00	−0.572799
$635$	13960.0	0.872418
$636$	0	0
$637$	10614.0	0.660192
$638$	−11160.0	−0.692521
$639$	0	0
$640$	−640.000	−0.0395285
$641$	−9240.00	−0.569357	−0.284679	−	0.958623i	$-0.591887\pi$
$641$	−9240.00	−0.569357	−0.284679	+	0.958623i	$0.591887\pi$
$642$	0	0
$643$	−17908.0	−1.09832	−0.549162	−	0.835716i	$-0.685053\pi$
$643$	−17908.0	−1.09832	−0.549162	+	0.835716i	$0.685053\pi$
$644$	936.000	0.0572726
$645$	0	0
$646$	−1176.00	−0.0716240
$647$	−7530.00	−0.457550	−0.228775	−	0.973479i	$-0.573472\pi$
$647$	−7530.00	−0.457550	−0.228775	+	0.973479i	$0.573472\pi$
$648$	0	0
$649$	−6120.00	−0.370156
$650$	3050.00	0.184047
$651$	0	0
$652$	−2692.00	−0.161698
$653$	22788.0	1.36564	0.682820	−	0.730586i	$-0.260753\pi$
$653$	22788.0	1.36564	0.682820	+	0.730586i	$0.260753\pi$
$654$	0	0
$655$	3540.00	0.211174
$656$	−6048.00	−0.359961
$657$	0	0
$658$	−3744.00	−0.221818
$659$	−16440.0	−0.971793	−0.485896	−	0.874016i	$-0.661507\pi$
$659$	−16440.0	−0.971793	−0.485896	+	0.874016i	$0.661507\pi$
$660$	0	0
$661$	−12421.0	−0.730894	−0.365447	−	0.930832i	$-0.619084\pi$
$661$	−12421.0	−0.730894	−0.365447	+	0.930832i	$0.619084\pi$
$662$	−1690.00	−0.0992201
$663$	0	0
$664$	816.000	0.0476912
$665$	3185.00	0.185728
$666$	0	0
$667$	−3348.00	−0.194355
$668$	14784.0	0.856303
$669$	0	0
$670$	1870.00	0.107827
$671$	−26310.0	−1.51369
$672$	0	0
$673$	6461.00	0.370064	0.185032	−	0.982732i	$-0.440761\pi$
$673$	6461.00	0.370064	0.185032	+	0.982732i	$0.440761\pi$
$674$	−17446.0	−0.997025
$675$	0	0
$676$	6096.00	0.346837
$677$	912.000	0.0517740	0.0258870	−	0.999665i	$-0.491759\pi$
$677$	912.000	0.0517740	0.0258870	+	0.999665i	$0.491759\pi$
$678$	0	0
$679$	3445.00	0.194708
$680$	480.000	0.0270694
$681$	0	0
$682$	9600.00	0.539007
$683$	14442.0	0.809089	0.404544	−	0.914518i	$-0.367430\pi$
$683$	14442.0	0.809089	0.404544	+	0.914518i	$0.367430\pi$
$684$	0	0
$685$	8430.00	0.470210
$686$	−13442.0	−0.748131
$687$	0	0
$688$	−4288.00	−0.237614
$689$	34770.0	1.92254
$690$	0	0
$691$	1892.00	0.104161	0.0520804	−	0.998643i	$-0.483415\pi$
$691$	1892.00	0.104161	0.0520804	+	0.998643i	$0.483415\pi$
$692$	−12528.0	−0.688213
$693$	0	0
$694$	18036.0	0.986509
$695$	−1535.00	−0.0837782
$696$	0	0
$697$	4536.00	0.246504
$698$	−11518.0	−0.624588
$699$	0	0
$700$	−1300.00	−0.0701934
$701$	7914.00	0.426402	0.213201	−	0.977008i	$-0.431611\pi$
$701$	7914.00	0.426402	0.213201	+	0.977008i	$0.431611\pi$
$702$	0	0
$703$	4459.00	0.239224
$704$	1920.00	0.102788
$705$	0	0
$706$	−11544.0	−0.615388
$707$	−16224.0	−0.863036
$708$	0	0
$709$	−1291.00	−0.0683844	−0.0341922	−	0.999415i	$-0.510886\pi$
$709$	−1291.00	−0.0683844	−0.0341922	+	0.999415i	$0.510886\pi$
$710$	−6060.00	−0.320321
$711$	0	0
$712$	7872.00	0.414348
$713$	2880.00	0.151272
$714$	0	0
$715$	−9150.00	−0.478588
$716$	−2040.00	−0.106478
$717$	0	0
$718$	−4092.00	−0.212691
$719$	30210.0	1.56696	0.783479	−	0.621418i	$-0.213443\pi$
$719$	30210.0	1.56696	0.783479	+	0.621418i	$0.213443\pi$
$720$	0	0
$721$	15925.0	0.822577
$722$	8916.00	0.459583
$723$	0	0
$724$	−4348.00	−0.223194
$725$	4650.00	0.238202
$726$	0	0
$727$	15680.0	0.799916	0.399958	−	0.916533i	$-0.369025\pi$
$727$	15680.0	0.799916	0.399958	+	0.916533i	$0.369025\pi$
$728$	−6344.00	−0.322973
$729$	0	0
$730$	−4310.00	−0.218521
$731$	3216.00	0.162720
$732$	0	0
$733$	13898.0	0.700320	0.350160	−	0.936690i	$-0.386127\pi$
$733$	13898.0	0.700320	0.350160	+	0.936690i	$0.386127\pi$
$734$	2138.00	0.107514
$735$	0	0
$736$	576.000	0.0288473
$737$	−5610.00	−0.280389
$738$	0	0
$739$	35300.0	1.75715	0.878573	−	0.477607i	$-0.158496\pi$
$739$	35300.0	1.75715	0.878573	+	0.477607i	$0.158496\pi$
$740$	−1820.00	−0.0904115
$741$	0	0
$742$	−14820.0	−0.733234
$743$	−28188.0	−1.39181	−0.695907	−	0.718132i	$-0.744997\pi$
$743$	−28188.0	−1.39181	−0.695907	+	0.718132i	$0.744997\pi$
$744$	0	0
$745$	9060.00	0.445547
$746$	−14266.0	−0.700155
$747$	0	0
$748$	−1440.00	−0.0703899
$749$	−1014.00	−0.0494670
$750$	0	0
$751$	25163.0	1.22265	0.611326	−	0.791379i	$-0.290637\pi$
$751$	25163.0	1.22265	0.611326	+	0.791379i	$0.290637\pi$
$752$	−2304.00	−0.111726
$753$	0	0
$754$	22692.0	1.09601
$755$	1015.00	0.0489267
$756$	0	0
$757$	7979.00	0.383093	0.191547	−	0.981484i	$-0.438650\pi$
$757$	7979.00	0.383093	0.191547	+	0.981484i	$0.438650\pi$
$758$	17114.0	0.820064
$759$	0	0
$760$	1960.00	0.0935483
$761$	−26622.0	−1.26813	−0.634065	−	0.773280i	$-0.718615\pi$
$761$	−26622.0	−1.26813	−0.634065	+	0.773280i	$0.718615\pi$
$762$	0	0
$763$	−28574.0	−1.35576
$764$	16224.0	0.768277
$765$	0	0
$766$	28656.0	1.35168
$767$	12444.0	0.585824
$768$	0	0
$769$	−35413.0	−1.66063	−0.830316	−	0.557293i	$-0.811840\pi$
$769$	−35413.0	−1.66063	−0.830316	+	0.557293i	$0.811840\pi$
$770$	3900.00	0.182528
$771$	0	0
$772$	1892.00	0.0882054
$773$	33834.0	1.57429	0.787144	−	0.616769i	$-0.211559\pi$
$773$	33834.0	1.57429	0.787144	+	0.616769i	$0.211559\pi$
$774$	0	0
$775$	−4000.00	−0.185399
$776$	2120.00	0.0980716
$777$	0	0
$778$	27000.0	1.24421
$779$	18522.0	0.851886
$780$	0	0
$781$	18180.0	0.832947
$782$	−432.000	−0.0197548
$783$	0	0
$784$	−2784.00	−0.126822
$785$	−1070.00	−0.0486496
$786$	0	0
$787$	13469.0	0.610061	0.305030	−	0.952343i	$-0.401333\pi$
$787$	13469.0	0.610061	0.305030	+	0.952343i	$0.401333\pi$
$788$	−10224.0	−0.462202
$789$	0	0
$790$	−11510.0	−0.518364
$791$	25818.0	1.16053
$792$	0	0
$793$	53497.0	2.39563
$794$	−2668.00	−0.119249
$795$	0	0
$796$	−11692.0	−0.520618
$797$	21168.0	0.940789	0.470395	−	0.882456i	$-0.344112\pi$
$797$	21168.0	0.940789	0.470395	+	0.882456i	$0.344112\pi$
$798$	0	0
$799$	1728.00	0.0765109
$800$	−800.000	−0.0353553
$801$	0	0
$802$	6948.00	0.305913
$803$	12930.0	0.568231
$804$	0	0
$805$	1170.00	0.0512262
$806$	−19520.0	−0.853055
$807$	0	0
$808$	−9984.00	−0.434698
$809$	−408.000	−0.0177312	−0.00886558	−	0.999961i	$-0.502822\pi$
$809$	−408.000	−0.0177312	−0.00886558	+	0.999961i	$0.502822\pi$
$810$	0	0
$811$	−36916.0	−1.59839	−0.799196	−	0.601070i	$-0.794741\pi$
$811$	−36916.0	−1.59839	−0.799196	+	0.601070i	$0.794741\pi$
$812$	−9672.00	−0.418006
$813$	0	0
$814$	5460.00	0.235102
$815$	−3365.00	−0.144627
$816$	0	0
$817$	13132.0	0.562338
$818$	−1138.00	−0.0486421
$819$	0	0
$820$	−7560.00	−0.321959
$821$	24636.0	1.04726	0.523631	−	0.851945i	$-0.324577\pi$
$821$	24636.0	1.04726	0.523631	+	0.851945i	$0.324577\pi$
$822$	0	0
$823$	−18187.0	−0.770303	−0.385151	−	0.922853i	$-0.625851\pi$
$823$	−18187.0	−0.770303	−0.385151	+	0.922853i	$0.625851\pi$
$824$	9800.00	0.414319
$825$	0	0
$826$	−5304.00	−0.223426
$827$	−1464.00	−0.0615578	−0.0307789	−	0.999526i	$-0.509799\pi$
$827$	−1464.00	−0.0615578	−0.0307789	+	0.999526i	$0.509799\pi$
$828$	0	0
$829$	−12295.0	−0.515106	−0.257553	−	0.966264i	$-0.582916\pi$
$829$	−12295.0	−0.515106	−0.257553	+	0.966264i	$0.582916\pi$
$830$	1020.00	0.0426563
$831$	0	0
$832$	−3904.00	−0.162676
$833$	2088.00	0.0868486
$834$	0	0
$835$	18480.0	0.765900
$836$	−5880.00	−0.243258
$837$	0	0
$838$	−18264.0	−0.752887
$839$	−46884.0	−1.92922	−0.964610	−	0.263681i	$-0.915063\pi$
$839$	−46884.0	−1.92922	−0.964610	+	0.263681i	$0.915063\pi$
$840$	0	0
$841$	10207.0	0.418508
$842$	5942.00	0.243201
$843$	0	0
$844$	−12700.0	−0.517953
$845$	7620.00	0.310220
$846$	0	0
$847$	5603.00	0.227298
$848$	−9120.00	−0.369318
$849$	0	0
$850$	600.000	0.0242116
$851$	1638.00	0.0659811
$852$	0	0
$853$	12197.0	0.489587	0.244793	−	0.969575i	$-0.421280\pi$
$853$	12197.0	0.489587	0.244793	+	0.969575i	$0.421280\pi$
$854$	−22802.0	−0.913663
$855$	0	0
$856$	−624.000	−0.0249157
$857$	28182.0	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$857$	28182.0	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$858$	0	0
$859$	25433.0	1.01020	0.505101	−	0.863060i	$-0.331455\pi$
$859$	25433.0	1.01020	0.505101	+	0.863060i	$0.331455\pi$
$860$	−5360.00	−0.212528
$861$	0	0
$862$	24084.0	0.951629
$863$	16968.0	0.669290	0.334645	−	0.942344i	$-0.391384\pi$
$863$	16968.0	0.669290	0.334645	+	0.942344i	$0.391384\pi$
$864$	0	0
$865$	−15660.0	−0.615556
$866$	17132.0	0.672251
$867$	0	0
$868$	8320.00	0.325345
$869$	34530.0	1.34793
$870$	0	0
$871$	11407.0	0.443756
$872$	−17584.0	−0.682878
$873$	0	0
$874$	−1764.00	−0.0682702
$875$	−1625.00	−0.0627829
$876$	0	0
$877$	−22423.0	−0.863365	−0.431682	−	0.902026i	$-0.642080\pi$
$877$	−22423.0	−0.863365	−0.431682	+	0.902026i	$0.642080\pi$
$878$	−14800.0	−0.568879
$879$	0	0
$880$	2400.00	0.0919363
$881$	−8442.00	−0.322836	−0.161418	−	0.986886i	$-0.551607\pi$
$881$	−8442.00	−0.322836	−0.161418	+	0.986886i	$0.551607\pi$
$882$	0	0
$883$	41207.0	1.57047	0.785236	−	0.619197i	$-0.212542\pi$
$883$	41207.0	1.57047	0.785236	+	0.619197i	$0.212542\pi$
$884$	2928.00	0.111402
$885$	0	0
$886$	−5160.00	−0.195659
$887$	−8472.00	−0.320701	−0.160351	−	0.987060i	$-0.551262\pi$
$887$	−8472.00	−0.320701	−0.160351	+	0.987060i	$0.551262\pi$
$888$	0	0
$889$	−36296.0	−1.36932
$890$	9840.00	0.370604
$891$	0	0
$892$	−8704.00	−0.326717
$893$	7056.00	0.264412
$894$	0	0
$895$	−2550.00	−0.0952370
$896$	1664.00	0.0620428
$897$	0	0
$898$	26400.0	0.981046
$899$	−29760.0	−1.10406
$900$	0	0
$901$	6840.00	0.252912
$902$	22680.0	0.837208
$903$	0	0
$904$	15888.0	0.584543
$905$	−5435.00	−0.199630
$906$	0	0
$907$	−21799.0	−0.798042	−0.399021	−	0.916942i	$-0.630650\pi$
$907$	−21799.0	−0.798042	−0.399021	+	0.916942i	$0.630650\pi$
$908$	15336.0	0.560510
$909$	0	0
$910$	−7930.00	−0.288876
$911$	−23544.0	−0.856254	−0.428127	−	0.903719i	$-0.640826\pi$
$911$	−23544.0	−0.856254	−0.428127	+	0.903719i	$0.640826\pi$
$912$	0	0
$913$	−3060.00	−0.110921
$914$	−36076.0	−1.30557
$915$	0	0
$916$	−12808.0	−0.461996
$917$	−9204.00	−0.331453
$918$	0	0
$919$	11072.0	0.397423	0.198711	−	0.980058i	$-0.436324\pi$
$919$	11072.0	0.397423	0.198711	+	0.980058i	$0.436324\pi$
$920$	720.000	0.0258018
$921$	0	0
$922$	11088.0	0.396056
$923$	−36966.0	−1.31826
$924$	0	0
$925$	−2275.00	−0.0808665
$926$	34274.0	1.21632
$927$	0	0
$928$	−5952.00	−0.210543
$929$	21654.0	0.764741	0.382371	−	0.924009i	$-0.375108\pi$
$929$	21654.0	0.764741	0.382371	+	0.924009i	$0.375108\pi$
$930$	0	0
$931$	8526.00	0.300138
$932$	−16608.0	−0.583705
$933$	0	0
$934$	−31776.0	−1.11321
$935$	−1800.00	−0.0629586
$936$	0	0
$937$	−42835.0	−1.49345	−0.746723	−	0.665135i	$-0.768374\pi$
$937$	−42835.0	−1.49345	−0.746723	+	0.665135i	$0.768374\pi$
$938$	−4862.00	−0.169243
$939$	0	0
$940$	−2880.00	−0.0999311
$941$	48534.0	1.68136	0.840682	−	0.541529i	$-0.182155\pi$
$941$	48534.0	1.68136	0.840682	+	0.541529i	$0.182155\pi$
$942$	0	0
$943$	6804.00	0.234962
$944$	−3264.00	−0.112536
$945$	0	0
$946$	16080.0	0.552649
$947$	−14676.0	−0.503597	−0.251798	−	0.967780i	$-0.581022\pi$
$947$	−14676.0	−0.503597	−0.251798	+	0.967780i	$0.581022\pi$
$948$	0	0
$949$	−26291.0	−0.899307
$950$	2450.00	0.0836721
$951$	0	0
$952$	−1248.00	−0.0424873
$953$	−27372.0	−0.930395	−0.465197	−	0.885207i	$-0.654017\pi$
$953$	−27372.0	−0.930395	−0.465197	+	0.885207i	$0.654017\pi$
$954$	0	0
$955$	20280.0	0.687168
$956$	21864.0	0.739678
$957$	0	0
$958$	7884.00	0.265888
$959$	−21918.0	−0.738028
$960$	0	0
$961$	−4191.00	−0.140680
$962$	−11102.0	−0.372082
$963$	0	0
$964$	−3772.00	−0.126025
$965$	2365.00	0.0788933
$966$	0	0
$967$	3581.00	0.119087	0.0595435	−	0.998226i	$-0.481035\pi$
$967$	3581.00	0.119087	0.0595435	+	0.998226i	$0.481035\pi$
$968$	3448.00	0.114486
$969$	0	0
$970$	2650.00	0.0877179
$971$	1824.00	0.0602832	0.0301416	−	0.999546i	$-0.490404\pi$
$971$	1824.00	0.0602832	0.0301416	+	0.999546i	$0.490404\pi$
$972$	0	0
$973$	3991.00	0.131496
$974$	−2758.00	−0.0907310
$975$	0	0
$976$	−14032.0	−0.460198
$977$	29778.0	0.975110	0.487555	−	0.873092i	$-0.337889\pi$
$977$	29778.0	0.975110	0.487555	+	0.873092i	$0.337889\pi$
$978$	0	0
$979$	−29520.0	−0.963701
$980$	−3480.00	−0.113433
$981$	0	0
$982$	28428.0	0.923802
$983$	−9402.00	−0.305063	−0.152532	−	0.988299i	$-0.548743\pi$
$983$	−9402.00	−0.305063	−0.152532	+	0.988299i	$0.548743\pi$
$984$	0	0
$985$	−12780.0	−0.413406
$986$	4464.00	0.144181
$987$	0	0
$988$	11956.0	0.384991
$989$	4824.00	0.155100
$990$	0	0
$991$	−24907.0	−0.798382	−0.399191	−	0.916868i	$-0.630709\pi$
$991$	−24907.0	−0.798382	−0.399191	+	0.916868i	$0.630709\pi$
$992$	5120.00	0.163871
$993$	0	0
$994$	15756.0	0.502767
$995$	−14615.0	−0.465655
$996$	0	0
$997$	27830.0	0.884037	0.442019	−	0.897006i	$-0.354263\pi$
$997$	27830.0	0.884037	0.442019	+	0.897006i	$0.354263\pi$
$998$	−19984.0	−0.633850
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.4.a.d.1.1	✓	1
3.2	odd	2		270.4.a.h.1.1	yes	1
4.3	odd	2		2160.4.a.q.1.1		1
5.2	odd	4		1350.4.c.o.649.1		2
5.3	odd	4		1350.4.c.o.649.2		2
5.4	even	2		1350.4.a.w.1.1		1
9.2	odd	6		810.4.e.j.271.1		2
9.4	even	3		810.4.e.r.541.1		2
9.5	odd	6		810.4.e.j.541.1		2
9.7	even	3		810.4.e.r.271.1		2
12.11	even	2		2160.4.a.g.1.1		1
15.2	even	4		1350.4.c.f.649.2		2
15.8	even	4		1350.4.c.f.649.1		2
15.14	odd	2		1350.4.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.d.1.1	✓	1	1.1	even	1	trivial
270.4.a.h.1.1	yes	1	3.2	odd	2
810.4.e.j.271.1		2	9.2	odd	6
810.4.e.j.541.1		2	9.5	odd	6
810.4.e.r.271.1		2	9.7	even	3
810.4.e.r.541.1		2	9.4	even	3
1350.4.a.i.1.1		1	15.14	odd	2
1350.4.a.w.1.1		1	5.4	even	2
1350.4.c.f.649.1		2	15.8	even	4
1350.4.c.f.649.2		2	15.2	even	4
1350.4.c.o.649.1		2	5.2	odd	4
1350.4.c.o.649.2		2	5.3	odd	4
2160.4.a.g.1.1		1	12.11	even	2
2160.4.a.q.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	270.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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